Pathwise Identification of the Memory Function of the Multifractional Brownian Motion with Application to Finance

We extend and adapt a class of estimators of the parameter H of the fractional Brownian motion in order to estimate the (time-dependent) memory function of a multifractional process. We provide: (a) the estimator's distribution when H in (0, 3/4); (b) the confidence interval under the null hypothesis H=1/2; (c) a scaling law, independent on the value of H, discriminating between fractional and multifractional processes. Furthermore, assuming as a model for the price process the multifractional Brownian motion, empirical evidence is offered which is able to conciliate the inconsistent results achieved in estimating the intensity of dependence in financial time series